What ways are there to really anger my PC's -- give them that sense of unbridled rage and fury -- without pissing the players off too?

I'd like to accomplish this through the story or in specific encounters. I don't want to just impose penalties or take away items or anything cheesy like that. I want the PC's to be throwing caution to the wind, busting down doors, attacking anything that moves, etc.

8 Answers
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I see a lot of insightful comments in the answers here; but, I think that there is a key element that hasn't been explicitly.

PCs get angry when something that they like is threatened.

Players get annoyed when something that they like is taken away.

The best way to engage your players is to threaten something dear to their characters, and give them a chance to earn it back (or protect it). The player needs to have a sense of control over what happens to their character.

Some of the other answers have hinted at this; no one's said it outright. Instill in them a sense of fear, but don't make them helpless, and reward them for rising to the challenge that the plot presents them with.

A story emerges when challenges face the characters, exposing flaws and weaknesses, and we see how the characters work to overcome these adversities. If the character doesn't have a role to play and a sense of control, the player will be more likely to respond poorly.

Really getting people in emotional place IC involves a bit of currency with the players and a little bit of trust. I have a few rules of thumb I always go for first when creating situations:

Plot and punishment are not the same thing.

I could just be sensitive about this but sometimes when a GM gives me plot by destroying something my character has worked at for a time I feel more like I'm being told to shut up and not ask for plot so be careful. This is especially if you haven't given focus to a player yet and you feel as if you need to bring the camera back to them; always go for complications that add rather than complications that subtract. Since you're a GM you're always effectively speaking.

If, for example, you were to attack their sister mercilessly while they were away you're telling the players "Leave your NPCs at your own peril" and you can expect to have giant parties wandering around with the PCs in no time. You're also saying "When you ask for something to do, I will make a problem and you will fix it." We as gamers like fixing problems usually. However, it doesn't take too long for that to become a fear that the GMs just going to kick over your lego tower and lack of concern to putting any work into anything. Always consider yourself to be making a statement about the nature of the game.

Fresh is always better than old.

You know what's awesome? Finding a new situation and handling it. It makes players feel empowered and the after effect on the world gives a sense of accomplishment. Erring towards players having problems and complications come to them rather than them happening upon problems at first while you're establishing your GMing relationship is always a strong move. In that sister example, if his sister ran up to the PC saying that someone beat her up and she needed help - we're also diluting the feel that bad things will happen while the PCs away and they're powerless without it because the PC is being brought into it. It's like coming home to a messy house and knowing you have a lot of chores to do; kind of takes the wind out of your sails. You absolutely don't want to give the player the feel that if they go off on adventures bad stuff will always happen to their stuff at home. It effectively starts to feel like you get punished for having a plot or punished for having fun. Unless they're really into it or you have a lot of currency, I don't suggest it. Start with external pressure, move to internal pressure, and if you've got a good relationship THEN twist the knife with helplessness if you want.

Players will tell you how they get emotional.

I'm kind of a Burning Wheel fangirl but players really, really do tell you what they care about and that's their first cue. Give them downtime and listen to where they go, what they do, and other things like that. Get a sense of where their values are before you begin so you know you can really go for it. Ask them OOCly what really annoys them and start a b**ch fest and then listen to them. They'll usually forget and frankly be really happy you were listening so keenly to them 9 times out of 10 if/when they recognize the situation.

Since they're in a dream world and if you have currency, my suggestion is to make the PCs feel impotent. Have someone else come through and be the hero and get the praise after they do the work. Find out what each of the players really enjoys and set up the situation and then right at the pay off of praise or reward have the camera stolen away to THAT STUPID GUY. It gets the party all together in frustration and upset averse to polarizing them and everyone has a tendency to naturally dislike things that are a little too perfect.

Speaking of things that are a little too perfect that's another situation wherein you can operate on subconscious player issues without making them upset. We tend to dislike things that seem fake. Play with that. Have everyone smile. Constantly smile. Shallow, cheerful statements no matter what. No. Matter. What. And then have something bad happen and let the NPCs smile and cheerful away. Again, injustice, frustration but it's also external.

Funny you mention Burning Wheel. Last night my BW character witnessed his mentor being slowly tortured to death in front of him. One of his motivations is seeking out and learning from magicians. He went completely unhinged. As a player I was happy as a clam, because now my character could get into all kinds of interesting trouble. But my character was torqued beyond belief.
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Erik SchmidtMay 19 '12 at 18:12

This is a small thing, but I don't think it has been specifically stated yet:

There are some things that you can take from the players, and some that you can't.

In our group, I know that we have lost money, items, and NPCs and been (as players) completely fine with it. But we had a guest DM for a few sessions, and they created enemies that caused the PCs to permanently lose experience - that was a line we were not cool with crossing.

One situation I found which worked quite well was a foe that would use fight and fade tactics. They would ambush the PCs and then retreat before the PCs could react. Each individual attack was minor, but this tactic was repeated over and over again, wearing down the PCs before they came across a big encounter where they got beat up because they had used up their spells and healing long before from the constant harassment.

I think the best way to anger the PCs without angering them is to do stuff to things they care about, rather than them. That way they aren't going to fell that the universe, and by extension you, is picking on them. Anger them by having antagonists trample over something they really care about.

Do your PCs have family? Have an antagonist kidnap or kill them. Or, more subtly, have an antagonist destroy their lives (by throwing them out of their home, for example). When the player finds out, this is likely to upset them.

Your players may not have a well developed backstory, in which case you'll have to work a bit harder. Is your player a paladin? Have the antagonist defile their God's temple. A hoarding thief? Steal their stash. Someone with a really nifty or signature weapon? Steal or break it (as long as there's some prospect of repair, the latter probably won't rile them, or the antagonist could have the only equivalent widget, which may have been their motivation in destroying it.

My point is (a) think about what they value (b) be creative (c) make sure that if you're taking away from them that you're ready to give back too - it may not be necessary immediately, but there needs to be a sense of progress as a result of the storyline you're planning.

Speaking personally, I love it when the GM is mean to me, so long as it creates story. If your players are sufficiently mature they will hopefully realise that what you're doing is good, but it's your responsibility to make good game out of it.

Know both the players at the table, and the characters in the adventure. Most good characters have strong motivations. Play to them. If they fight injustice, create an injustice. If they crave wealth, have a rival enter the story and gloat about how they just made a big score.

Another way is to create memorable NPCs and then drastically alter the circumstances surrounding them. Have the stable boy that has been caring for the PCs horses get beat up. Have the wise cracking barmaid with the heart of gold get an incurable illness. Have the innkeeper of their favourite inn sell it to a real jerk with a horrible attitude toward adventurers.

Players get angry when they feel that they are being treated unfairly. Try to keep the circumstances causing the character rage within the arena of the story. Taking away the barbarian's prized belt of strength will anger both the character and the player. However, having an evil wizard curse the barbarian's lover with incurable warts all over her entire body will not anger the player, but some serious barbarian wrath will be created.

Often the easiest way to get a character angry while also generating player interest is simply through great dialogue between PCs and NPCs. This can result in some blood-boiling anger from the characters and roleplaying opportunities for the players.

Our halfling ranger had chatted up a girl in a town we had just been in. Our quest goal was to conquer a castle that the Iron Circle had taken over. We ended up on top of the castle wall and made perception checks at the door outside every turret. What we heard made our halfling quite angry (lets just say it was about the girl and leave it at that) to the point where he would break down the door and fire an arrow into the first guard he saw. Our DM let the halfling's rage jack the initiative order and it was hilarious.

It was a fun way to lightheartedly add a little PC rage to the encounter without making the players angry.

Wow, there's a lot of great comments already... I'll just add my 2 cents because I try and accomplish this quite a lot.

The first thing you have to have is players who can handle it... without this, it gets personal and ugly.

What I tend to do is understand the players and have them document goals.. I use goal related reward bonuses, so I have an idea of what's really important to them.

I use socio-political events to create a rich background for the campaign, with recurring NPC's who take a hand in thwarting PC's or aiding them (matching their goals).

There's a few in the group who are more power gamers and taking "things" is the worst that can happen to them... either outright or hinting that they were close and something happened to prevent them from getting the item.

Even when PC's succeed, I try and make sure they aware of the cost... in wounds (temp or perm loss to their characters), human life, tragedy, economically. That's especially powerful for the story oriented players...

I never let them forget that ultimately, they are out there beating others down and taking their stuff... good and evil are sometimes very relative and I make them experience that fully.

I try not to upset characters directly - i.e. I never penalize them or take away key items or abilities (unless there is a way to get it back and it does not undercut them in the upcoming tasks).

Honestly, my players tell me everything I give their character comes with a price.... and that's just the kind of gritty element that suits our group. It may not be for everyone!